Methods, devices and systems for separating overlappingly transmitted signals and enabling joint spectrum access

ABSTRACT

In one example embodiment, a device includes a memory configured to store computer-readable instructions therein and a processor. The processor is configured to execute the computer-readable instructions to receive a mixture signal, and determine a first signal and a second signal from the mixture signal, the first signal being a signal of a first technology and the second signal being a signal of a second technology, the first and second signals being overlappingly transmitted signals, at least one of the first signal and the second signal being used for processing of information associated with the at least one of the first signal and the second signal.

CROSS-REFERENCE TO RELATED APPLICATIONS

This This application is a continuation of U.S. patent application Ser.No. 14/984,160 filed on Dec. 30, 2015 (U.S. Pat. No. 9,942,774), whichis a continuation of U.S. patent application Ser. No. 14/804,499 filedon Jul. 21, 2015 (U.S. Pat. No. 9,277,418), the entire contents of whichare hereby incorporated by reference.

BACKGROUND

The radio-frequency (RF) electromagnetic spectrum, extending from below1 MHz to above 100 GHz, represents a finite resource that is shared byvariety of devices including devices operating using wirelesscommunications standards, radar devices, television broadcasts, radionavigation and other RF devices. The increasing demand by consumers forhigher data rates induces competition among RF devices for accessing thefinite RF spectrum. Accordingly, appropriate federal agencies haverecently recommended that 1000 MHz of federally-controlled RF spectrumshould be freed or shared with the private industry in order to meet theever growing need for wireless communications-based services.

A signal mixture represents a super-position of a plurality ofindividual signals with the addition of possible noise. Examples ofsignal mixtures occur in many applications. For example, when aplurality of RF devices such as radars and wireless communicationsdevices are simultaneously operating over the same frequency spectrum,the received baseband signal at each individual device is a signalmixture that is a superposition of the signals from each RF device. Ifsuch signal is used as an input or source into receivers at radars orwireless communication devices, the performance of the radars andwireless communications will degrade relative to their actualperformance potential when no super-position by external signals ispresent.

A special case of separation of mixtures which is more challenging areconvolutive mixtures which may arise in several signal processing fieldssuch as speech processing, music processing, sonar processing, radiocommunications processing, antenna array data processing, astronomicaldata processing, satellite imagery processing, functional brain imageprocessing, etc. For example, in acoustic processing, such mixturesarise due to time delays resulting from sound propagation over space andthe multipath induced from reflections of sound by different objects.

Radars are used for a variety of applications includingair-traffic-control, weather forecasting, automotive collision avoidancesystems, ground penetrating radars for finding underground resources,altimeters for elevation measurements, geophysical monitoring ofresources by synthetic aperture radar (SAR) systems, etc. Studies haveshown that the effect of wireless communications interference on radarsystems may severely inhibit the performance of radar devices/systems.Therefore, conventionally, when a primary device (e.g., a radar device)operates in a given spectrum (e.g., frequency band), secondary devicessuch as devices communicating using wireless communicationstechnologies, have not been allowed to operate in the given spectrum.

Various solutions have been proposed for enabling the use of “whitespectrum” (e.g., RF spectrum used by primary devices) by the secondarydevices. This means allowing secondary wireless devices to operate whenthe primary wireless device(s) are not active within a frequency bandand geographical area. One such proposed solution is referred to asDynamic Spectrum Access (DSA), with Dynamic Frequency Selection (DFS)being a particular example of the DSA solution.

Another proposed solution (not currently implemented or not implementedfor spectrum sharing purposes) might be radar systems such as passivesystems and multiple-input multiple-output (MIMO) radars to alleviatethe spectrum congestion problem and make more spectrum available for useby wireless communications systems. However these systems are much morecomplex than the existing deployed radar systems. Furthermore,replacements of existing radar systems may be cost prohibitive andconsequently such proposed systems are not currently feasible.

Therefore, more robust methods allowing for separation ofsimultaneously/overlappingly transmitted signals as well as simultaneousoperation of wireless communications and radar devices/systems aredesirable.

SUMMARY

Some example embodiments relate to methods, apparatuses and systems forenabling simultaneous operation of different technology based devicesover a shared spectrum.

In one example embodiment, a device includes a memory configured tostore computer-readable instructions therein and a processor. Theprocessor is configured to execute the computer-readable instructions toreceive a mixture signal, and determine a first signal and a secondsignal from the mixture signal, the first signal being a signal of afirst technology and the second signal being a signal of a secondtechnology, the first and second signals being overlappingly transmittedsignals, at least one of the first signal and the second signal beingused for processing of information associated with the at least one ofthe first signal and the second signal.

In yet another example embodiment, the overlapping transmission of thefirst signal and the second signal includes transmission of the firstsignal and the second signal over a shared spectrum.

In yet another example embodiment, the overlapping transmission of thefirst signal and the second signal includes a spatial overlap of thefirst signal and the second signal as well as overlaps of the firstsignal and the second signal in time and frequency domains.

In yet another example embodiment, the first technology is a radartechnology and the second technology is a wireless communicationsstandard.

In yet another example embodiment, the first technology and the secondtechnology are radar technologies.

In yet another example embodiment, the first technology and the secondtechnology are different wireless communications standards.

In yet another example embodiment, the mixture signal is a convolutivemixture of a first convolutive process, a second convolutive process anda noise signal, the first convolutive process corresponds to aconvolution of the first signal and a first impulse response of a firstsystem operating based on the first technology, and the secondconvolutive process corresponds to a convolution of the second signaland a second impulse response of a second system operating based on thesecond technology.

In yet another example embodiment, the processor is configured tominimize a cost function associated with the received mixture signal,the cost function being a function of the first signal and the secondsignal, and determine the first signal and the second signal from amongpossible sets of values of the first signal and the second signal thatminimize the cost function.

In yet another example embodiment, the processor is further configuredto minimize the cost the function based on an iterative process.

In yet another example embodiment, the cost function is based on thereceived mixture signal, the first impulse response, the second impulseresponse, a first regularization term corresponding to the first signaland a second regularization term corresponding to the second signal.

In yet another example embodiment, the noise signal is colored noisewith an associated power spectral density.

In yet another example embodiment, the processor is configured to adjustthe cost function based on the power spectral density of the colorednoise.

In yet another example embodiment, the mixture signal is a combinationof a convolutive process, a signal represented by combination of thesecond signal and a transform, and a noise signal, the convolutiveprocess corresponds to a convolution of the first signal and an impulseresponse of a system operating based on the first technology, thetransform is at least one of an undercomplete, complete and overcompletetransform.

In yet another example embodiment, the process is configured to minimizea cost function associated with the received mixture signal, the costfunction being a function of the first signal and the second signal, anddetermine the first signal and the second signal from among possiblesets of values of the first signal and the second signal that minimizethe cost function.

In yet another example embodiment, the processor is further configuredto minimize the cost function based on an iterative process.

In yet another example embodiment, the cost function is based on thereceived mixture signal, the impulse response, the transform, a firstregularization term corresponding to the first signal and a secondregularization term corresponding to the second signal.

In yet another example embodiment, a radar system includes a deviceconfigured to function as a receiver of radar signals for the radarsystem. The device includes a memory configured to storecomputer-readable instructions therein and a processor. The processor isconfigured to execute the computer-readable instructions to receive amixture signal, and determine a first signal and a second signal fromthe mixture signal, the first signal being a signal of a firsttechnology and the second signal being a signal of a second technology,the first and second signals being overlappingly transmitted signals, atleast one of the first signal and the second signal being used forprocessing of information associated with the at least one of the firstsignal and the second signal.

In yet another example embodiment, a wireless communications systemincludes a device configured to function as a receiver of signals forthe wireless communications system. The device includes a memoryconfigured to store computer-readable instructions therein and aprocessor. The processor is configured to execute the computer-readableinstructions to receive a mixture signal, and determine a first signaland a second signal from the mixture signal, the first signal being asignal of a first technology and the second signal being a signal of asecond technology, the first and second signals being overlappinglytransmitted signals, at least one of the first signal and the secondsignal being used for processing of information associated with the atleast one of the first signal and the second signal.

In yet another example embodiment, when the processor processes thefirst signal, the processor is configured to at least one of detectobjects and process a parameter, and when the processor processes thesecond signal, the processor is configured to establish wireless datacommunications between the receiver, a transmitter and other networkelements in a wireless data communications system operating based on thesecond technology.

In one example embodiment, a device includes a memory configured tostore computer-readable instructions therein and a processor. Theprocessor is configured to execute the computer-readable instructions toreceive a mixture signal, the mixture signal being a mixture of at leasttwo signals that are overlappingly transmitted. The processor is furtherconfigured to determine each of the at least two signals based on a costfunction associated with the mixture signal, and process at least one ofthe at least two signals transmitted by a corresponding transmitter tobe received by the device.

In yet another example embodiment, the processor is configured todetermine each of the at least two signals as a set of solutions thatminimize a cost function among all possible sets of values of each ofthe at least two signals.

In yet another example embodiment, the at least two signals aretransmitted by a single transmitter.

In yet another example embodiment, one of the at least two signals istransmitted by the single transmitter based on a first technology, andanother one of the at least two signals is transmitted by the signaltransmitter based on a second technology.

In yet another example embodiment, the overlapping transmission of theat least two signals includes transmission of the at least two signalsover a shared spectrum.

In yet another example embodiment, the overlapping transmission of theat least two signals includes a spatial overlap of the at least twosignals as well as overlaps of the at least two signals in time andfrequency domains.

In yet another example embodiment, each of the at least two signals aretransmitted based on one of at least two different technologies.

In yet another example embodiment, each of the at least two differenttechnologies correspond to at least one of a radar technology and awireless communications standard.

In yet another example embodiment, the processor processes the at leasttwo signals according to a functionality of a system to which the devicebelongs, the system being at least one of a radar based data collection,detection, imaging and tracking system, a wireless radio communicationssystem, a medical imaging system, and an acoustic signal processingsystem.

BRIEF DESCRIPTION OF THE DRAWINGS

Example embodiments will become more fully understood from the detaileddescription given herein below and the accompanying drawings, whereinlike elements are represented by like reference numerals, which aregiven by way of illustration only and thus are not limiting of thepresent disclosure, and wherein:

FIG. 1 illustrates a setting in which a wireless communications systemand a radar system operate simultaneously, according to an exampleembodiment;

FIG. 2 illustrates a setting in which two radar systems operatesimultaneously, according to an example embodiment;

FIG. 3 illustrates a receiver for receiving signals of the first systemshown in FIG. 1, according to an example embodiment;

FIG. 4 illustrates a receiver for receiving signals of the second systemshown in FIG. 1, according to an example embodiment;

FIG. 5 describes a method of determining data sources from a mixturesignal, according to an example embodiment;

FIG. 6 describes a method of determining data sources from a mixturesignal, according to an example embodiment;

FIG. 7 describes a method of determining data sources from a mixturesignal, according to an example embodiment; and

FIG. 8 describes a method of determining data sources from a mixturesignal, according to an example embodiment.

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS

Various embodiments will now be described more fully with reference tothe accompanying drawings. Like elements on the drawings are labeled bylike reference numerals.

Detailed illustrative embodiments are disclosed herein. However,specific structural and functional details disclosed herein are merelyrepresentative for purposes of describing example embodiments. Thisdisclosure may, however, be embodied in many alternate forms and shouldnot be construed as limited to only the embodiments set forth herein.

Accordingly, while example embodiments are capable of variousmodifications and alternative forms, the embodiments are shown by way ofexample in the drawings and will be described herein in detail. Itshould be understood, however, that there is no intent to limit exampleembodiments to the particular forms disclosed. On the contrary, exampleembodiments are to cover all modifications, equivalents, andalternatives falling within the scope of this disclosure. Like numbersrefer to like elements throughout the description of the figures.

Although the terms first, second, etc. may be used herein to describevarious elements, these elements should not be limited by these terms.These terms are only used to distinguish one element from another. Forexample, a first element could be termed a second element, andsimilarly, a second element could be termed a first element, withoutdeparting from the scope of this disclosure. As used herein, the term“and/or,” includes any and all combinations of one or more of theassociated listed items.

When an element is referred to as being “connected,” or “coupled,” toanother element, it can be directly connected or coupled to the otherelement or intervening elements may be present. By contrast, when anelement is referred to as being “directly connected,” or “directlycoupled,” to another element, there are no intervening elements present.Other words used to describe the relationship between elements should beinterpreted in a like fashion (e.g., “between,” versus “directlybetween,” “adjacent,” versus “directly adjacent,” etc.).

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting. As used herein, thesingular forms “a”, “an”, and “the” are intended to include the pluralforms as well, unless the context clearly indicates otherwise. It willbe further understood that the terms “comprises”, “comprising”,“includes” and/or “including”, when used herein, specify the presence ofstated features, steps, operations, elements, and/or components, but donot preclude the presence or addition of one or more other features,integers, steps, operations, elements, components, and/or groupsthereof.

It should also be noted that in some alternative implementations, thefunctions/acts noted may occur out of the order noted in the figures.For example, two figures shown in succession may in fact be executedsubstantially concurrently or may sometimes be executed in the reverseorder, depending upon the functionality/acts involved.

Specific details are provided in the following description to provide athorough understanding of example embodiments. However, it will beunderstood by one of ordinary skill in the art that example embodimentsmay be practiced without these specific details. For example, systemsmay be shown in block diagrams so as not to obscure the exampleembodiments in unnecessary detail. In other instances, well-knownprocesses, structures and techniques may be shown without unnecessarydetail in order to avoid obscuring example embodiments.

In the following description, illustrative embodiments will be describedwith reference to acts and symbolic representations of operations (e.g.,in the form of flow charts, flow diagrams, data flow diagrams, structurediagrams, block diagrams, etc.) that may be implemented as programmodules or functional processes include routines, programs, objects,components, data structures, etc., that perform particular tasks orimplement particular abstract data types and may be implemented usingexisting hardware at existing network elements. Such existing hardwaremay include, but is not limited to, one or more of Central ProcessingUnits (CPUs), Digital Signal Processors (DSPs), Graphical ProcessingUnits (GPUs), Very Large Scale Integration (VLSI) circuits,Application-Specific-Integrated-Circuits (ASICs), Field ProgrammableGate Arrays (FPGAs), computers or the like.

Although a flow chart may describe the operations as a sequentialprocess, many of the operations may be performed in parallel,concurrently or simultaneously. In addition, the order of the operationsmay be re-arranged. A process may be terminated when its operations arecompleted, but may also have additional steps not included in thefigure. A process may correspond to a method, function, procedure,subroutine, subprogram, etc. When a process corresponds to a function,its termination may correspond to a return of the function to thecalling function or the main function.

As disclosed herein, the term “storage medium” or “computer readablestorage medium” may represent one or more devices for storing data,including read only memory (ROM), random access memory (RAM), magneticRAM, core memory, magnetic disk storage mediums, optical storagemediums, flash memory devices and/or other tangible machine readablemediums for storing information. The term “computer-readable medium” mayinclude, but is not limited to, portable or fixed storage devices,optical storage devices, and various other mediums capable of storing,containing or carrying instruction(s) and/or data.

Furthermore, example embodiments may be implemented by hardware,software, firmware, middleware, microcode, hardware descriptionlanguages, or any combination thereof. When implemented in software,firmware, middleware, or microcode, the program code or code segments toperform the necessary tasks may be stored in a machine or computerreadable medium such as a computer readable storage medium. Whenimplemented in software, a processor or processors will perform thenecessary tasks.

A code segment may represent a procedure, function, subprogram, program,routine, subroutine, module, software package, class, or any combinationof instructions, data structures or program statements. A code segmentmay be coupled to another code segment or a hardware circuit by passingand/or receiving information, data, arguments, parameters or memorycontent. Information, arguments, parameters, data, etc. may be passed,forwarded, or transmitted via any suitable means including memorysharing, message passing, token passing, network transmission, etc.

Example embodiments described herein enable simultaneous operation ofdevices/systems of different technologies over a shared frequencyspectrum while detrimental interference of signals of one of thedifferent technologies on signals of another one of the differenttechnologies is minimized

Example embodiments described herein provide a signal processingapproach, in which a first device of a first technologydetermines/estimates a signal transmitted according to the firsttechnology and destined for the first device, from a mixed signalreceived at the first device. The mixed signal includes, among varioustypes of interference signals, a signal simultaneously transmittedaccording to a second technology over the same frequency spectrum (thefirst and second signals overlap in time and frequency domains). Thefirst device may then utilize the determined/estimated first signal forfurther processing of information associated with the first signal.

Example embodiments may be utilized in conjunction with various known orto be developed Wireless Local Area Network Technologies (WLANs).Furthermore, example embodiments may also be utilized in conjunctionwith Radio Access Networks (RANs) such as: Universal MobileTelecommunications System (UMTS); Global System for Mobilecommunications (GSM); Advance Mobile Phone Service (AMPS) system; theNarrowband AMPS system (NAMPS); the Total Access Communications System(TACS); the Personal Digital Cellular (PDC) system; the United StatesDigital Cellular (USDC) system; the code division multiple access (CDMA)system described in EIA/TIA IS-95; a High Rate Packet Data (HRPD)system, a High Rate Packet Data (HRPD) system, WorldwideInteroperability for Microwave Access (WiMAX); 4G Long Term Evolution(LTE); Wi-Fi; Ultra Mobile Broadband (UMB); and 3^(rd) GenerationPartnership Project LTE (3GPP LTE).

Example embodiments described herein enable simultaneous operation ofdevices/systems where the devices/systems are physically co-located anda mixture of signals transmitted by the devices/systems is composed ofmultiple convolutions and/or multiple signals that are effectivelyrepresented in a transform domain in order to allow reconstruction ofthe signals. In one example embodiment, the signals co-existsimultaneously in time and overlap in frequency.

Example embodiments described herein provide a signal processingapproach, in which a first device of a first technologydetermines/estimates a signal transmitted according to the firsttechnology and destined for the first device, from a mixture signalreceived at the first device. The mixture signal includes, among varioustypes of interference signals, a signal simultaneously transmittedaccording to a second technology over the same frequency spectrum (thefirst and second signals overlap in time and frequency domains). Thefirst device may then utilize the determined/estimated first signal forfurther processing of information associated with the first signal.

Furthermore, example embodiments described herein provide a signalprocessing approach, in which two or more systems operating based on thesame technology (e.g., two or more radar systems) that are co-located invicinity of one another (spatially overlap) may overlappingly transmitsignals (i.e., the signals may entirely or partially overlap in time andfrequency domains). Example embodiments allow for separation of theoverlappingly transmitted signals by a receiver of each of the two ormore systems and utilization of the separated signals for processinginformation associated with the separated signals according to theapplication/use of each of the two or more systems.

FIG. 1 illustrates a setting in which a wireless communications systemand a radar system operate simultaneously, according to an exampleembodiment. As shown in FIG. 1, in a setting 100 two different systemsco-exist (i.e., physically co-exist meaning that the two systems (and/orthe signals transmitted by the two systems) are in the geographicalvicinity of one another). The first system is the system 120 and thesecond system is the system 130. The systems 120 and 130 may operatebased on different technologies. In the example embodiment shown in FIG.1, the system 120 may be a radar system and the system 130 may be awireless communications system. However, example embodiments are notlimited to wireless communications and radar systems but may encompassany two systems operating according to different technologies. Forpurposes of describing example embodiments, the first system 120 and thesecond system 130 are considered to exist in vicinity of one another(i.e., the first system 120 and the second system 130 overlap spatiallysuch that when the first system 120 and the second 130 transmit signalson the same or overlapping frequencies and the same or overlappingtimes, the signals of each induce interference on signals of the otherof the two systems (each of the first system 120 and the second system130 experience degradation in their performance due to the transmittedsignal of the other of the two systems.))

The first system 120 may be a system that operates based on a differenttechnology than the technology based on which the second system 130operates. For example and as shown in FIG. 1, the first system 120 maybe a radar system. The radar system 120 may include a radar receiver122, a radar 124, and a radar object of detection 126. The radarreceiver 122 may control the operation of the radar 124, as will bedescribed below. The radar 124 may transmit a signal 128 to the radarobject of detection 126. The echo/reflection of the signal 128 may bethe signal 129 received back/detected by the radar 124 and processed bythe radar receiver 122. The radar object of detection 126 may be anytype of object/information to be detected, imaged, tracked, processed,and/or monitored by the radar 124.

The radar system 120 may be any coherent based radar system such as aweather radar system, surveillance radar system, airport traffic radarsystem, ground penetrating radar system, search and rescue radar system,car radar system including those with multiple array elements andmultiple antennas (MIMO). The radar may be in a staring mode, scanningmode, circling mode, stripmap mode, etc. The radar may also be in anyone of an imaging, tracking, detection or other modes.

The second system 130 may include components necessary for enablingcommunication according to the corresponding technology. For example, inFIG. 1 and assuming that the second system 130 operates according to awireless communications technology (e.g., GSM based wirelesscommunications system, CDMA based communications system, etc.), thesecond system 130 may include a wireless access point 132 communicatingwith communication client devices 134 (which may be hereinafter referredto as user equipment (UE)) via exchange of signals 136. The Wirelessaccess point 132 may differ from one wireless communications technologyto another but regardless of the underlying technology, enables the UEs134 to establish voice/data communication with other devices and/ornetwork components in the wireless communications system 130.

In one example embodiment and as shown in FIG. 1, the wireless accesspoint 132 may be a base station (e.g., macro cell base station, smallcell base station, femto cell base station, etc.). However, the exampleembodiments are not limited thereto but may encompass any other type ofaccess point through which the UEs 134 may establish voice/datacommunications with other UEs (in the same network or differentnetworks) or other network components. For example, the wireless accesspoint 132 may be a router, when the wireless communications system 120is a wireless local area network (WLAN) operating according to knownWLAN standards such as IEEE 802 standards. Furthermore, while somecomponents of the second system 130 are illustrated in FIG. 1, any othercomponent necessary for enabling wireless communication within thesecond system 130 is implicitly included (e.g., network access points,core network elements, etc.).

More generally, the first system 120 and the second system 130 may besystems of sensors and/or system of communication devices using the samespectrum resources where the waveforms may be electromagnetic, acousticor otherwise. The wireless communications and radar platforms may bestationary or moving on the ground, in the air/space or at the sea.

The radar system 120 may operate in one or more frequency bands (e.g., 5GHz band).

In one example embodiment, the signals 136 of the wirelesscommunications system 130 and the signals 128/129 of the radar system120 may be transmitted simultaneously over the same (or overlapping)frequency band/spectrum such that the signals 136 and 128/129 overlap intime and/or frequency domains (i.e., the signals 136 and 128/129 may besaid to share a spectrum, with the shared spectrum being associated withone or more specific frequencies such as 2 GHz, 5 GHz, etc.). Forexample, signals 136 and 128/129 may be transmitted over the entireand/or overlapping portions of the 5 GHz frequency band. Accordingly,the signals of each of the systems 120 and 130 (e.g., signals 136, 128and 129) may induce interference on signals of the other one of thesystems 120 and 130. The interference caused by each of the signals 136and 128/129 on the other one of the signals 136 and 128/129 isillustrated as interference signal 140 in FIG. 1.

For example, the signals 136 of the system 130 may interfere with thetransmitted and received signals 128/129 of the system 120. Accordingly,the mixed signal, as received at the radar receiver 122 may be a mixedsignal that is a combination of the signals 129 and 136 as well as apossible noise signal.

Similarly, the signals 128/129 of the system 120 may interfere with thesignals 136 of the system 130. Accordingly, the mixed signal, asreceived at a receiver of any one or more of the components in thesystem 130 (e.g., a receiver of any one of the UEs 134 and/or thewireless access point 132) may be a mixed signal that is a combinationof the signals 129 and 136 as well as a possible noise signal.

As will be described in greater detail below, example embodiments enablea receiver in each of the systems 120 and 130 to separate acorresponding one of the signals 129 and 136 from the mixture receivedat the receiver and perform further processing thereof in accordancewith the functionality of the device in each system at which thereceiver is located.

While FIG. 1 illustrates a setting in which only two systems (system 120and system 130) operating according to different technologies aredeployed, example embodiments are not limited thereto. For example,there may be a more than two systems deployed in the setting 100 each ofwhich operates based on a different technology and/or any pair of two ormore of the deployed systems may operate based on the same technologywhile at least one of the deployed system operates based on a differenttechnology. Regardless of the number of systems in the setting 100, eachsystem's transmitted signals may induce interference such asinterference signal 140 on the other systems in the setting 100 and thusthe signal received at a receiver in each of the co-existing systems maybe a mixture of overlappingly transmitted signals of each and every oneof the co-existing systems.

FIG. 2 illustrates a setting in which two systems operating based on thesame technology overlappingly transmit signals, according to an exampleembodiment. As shown in FIG. 2, in a setting 200 two different radarsystems co-exist (i.e., physically co-exist meaning that the two radarsystems (and/or signals transmitted by the two radar systems) are in thegeographical vicinity of one another). The first radar system is formedby the radar 240, the radar receiver 245 and the radar object ofdetection 250. The first radar system may transmit radar pulses 246toward the radar object of detection 250 and receive echoes 247 thereof.Similarly the second radar system is formed by the radar 260, the radarreceiver 265 and the radar object of detection 250. The second radarsystem may transmit radar pulses 266 toward the radar object ofdetection 250 and receive echoes 267 thereof. The frequency spectrumover which signals of the first and second radar system are transmitted,may overlap. Due to the overlap between the operational spectrum of theradar devices, as well as the radar systems' geographical proximity,each radar system's received signal 246 and 266 contains interference270 from the other one of the radar system. The interference in bothradar systems may come through the antenna mainlobe, side-lobe orback-lobe.

While FIG. 2 has been described with reference to two radar systems,example embodiments are not limited thereto and inventive concepts maybe applied to any two or more systems operating based on the sametechnology.

FIG. 3 illustrates a receiver for receiving signals of the first systemshown in FIG. 1, according to an example embodiment. In the exampleembodiment described above with reference to FIG. 1, the first system120 is described as a radar system. However, as mentioned, the firstsystem 120 is not limited to a radar system.

The receiver of FIG. 3 may be the radar receiver 122 of the first system120 of FIG. 1 that is to receive a radar signal transmitted by the radar124 to the radar object of detection 126 and reflected back to the radar124 from the radar object of detection 126.

As shown in FIG. 3, the radar receiver 122 may include a storage mediumdevice 345 and a processor 350. While FIG. 3 illustrates the radarreceiver 122 as including three components, example embodiments are notlimited thereto and the radar receiver 122 may include any number ofadditional components necessary for performing various functions withinthe radar system 120.

The storage medium device 345 may store, among other information, a setof computer-readable instructions and parameters for determining asignal of the first system 120 transmitted to the radar receiver 122 inpresence of interference induced by the signal 140 described above withreference to FIG. 1, as will be described below.

The processor 350 may execute the set of computer-readable instructionsfor performing the functions necessary to determine a signal of thefirst system 120 transmitted to the radar receiver 122, as will bedescribed below. Accordingly, the execution of the computer-readableinstructions by the processor 350 may transform the processor 350 into aspecial purpose processor for performing the underlying functions. Inaddition to determining the signal of the first system 120, theprocessor 350 may further execute additional computer-readableinstructions for processing information associated with the receivedsignal, as will be further described below.

FIG. 4 illustrates a receiver for receiving signals of the second systemshown in FIG. 1, according to an example embodiment. In the exampleembodiment described above with reference to FIG. 1, the second system130 is described as a wireless communications system. However, asmentioned, the second system 130 is not limited to a wirelesscommunications system.

The receiver 450 shown in FIG. 4 may be a receiver at any one of thecomponents in the second system 130 of FIG. 1 that is to receive asignal transmitted according to the technology based on which the secondsystem 130 operates. For example, the receiver 450 shown in FIG. 3 maybe a receiver at the UE 134, a receiver at the wireless access node 132or a receiver at any other network component within the second system130.

As shown in FIG. 4, the receiver 450 may include a storage medium device455, a processor 460 and an antenna 465. While FIG. 4 illustrates thereceiver 450 as including three components, example embodiments are notlimited thereto and the receiver 450 may include any number ofadditional components necessary for performing various functions withinthe second system 130.

The storage medium device 455 may store, among other information, a setof computer-readable instructions and parameters for determining asignal of the second system 130 transmitted to the receiver 450, as willbe described below.

The processor 460 may execute the set of computer-readable instructionsfor performing the functions necessary to determine a signal of thesecond system 130 transmitted to the receiver 450, as will be describedbelow. Accordingly, the execution of the computer-readable instructionsby the processor 460 may transform the processor 460 into a specialpurpose processor for performing the underlying functions. In additionto determining the signal of the second system 130 in presence ofinterference signal 140 described above with reference to FIG. 1, theprocessor 450 may further execute additional computer-readableinstructions for processing information associated with the receivedsignal, as will be further described below.

The antenna 465 may be any known or to be developed antennainstalled/incorporated into the receiver 450 (which may vary dependingon the component of the second system 130 in which the receiver 450 isembedded). The antenna 465 may be used to receive signals (which may bea mixture of the signal of the second system 130 as well as interferenceinduced by an overlappingly transmitted signal of the first system 120(in the form of interference signal 140 discussed above with referenceto FIG. 1, as well as additional noise interference)). The antenna 465may additionally be used to transmit data/information/signals to othercomponents of the second system 130 (e.g., the antenna 465 may be atransceiver antenna).

Hereinafter, example embodiments for determining each of the signalsfrom a mixture signal received at a receiver, will be described withreference to FIGS. 5-8. The methods described with reference to FIGS.5-8 may be implemented at any receiver of any system such as thosedescribed above. For purposes of discussion, FIGS. 5-8 will be describedwith reference to the radar receiver 122 shown in FIG. 1. Furthermore,throughout FIGS. 5-8, the signals to be determined/extracted from thereceived mixture signal (i.e., first and second signals) may also bereferred to as data sources.

FIG. 5 describes a method of determining data sources from a mixturesignal, according to an example embodiment.

At S500, the radar receiver 122 (via the processor 350 associatedtherewith) receives a mixture signal. The mixture signal may beconvolutive mixture that is a combination of two convolutive processes(e.g., (h₁*x₁)(n) and (h₂*x₂)(n), each being a different convolutiveprocess in Equation (1)). The mixture signal may also include randomnoise. The convolutive mixture signal y is as shown below:y(n)=(h ₁ *x ₁)(n)+(h ₂ *x ₂)(n)+w(n)   (1)where h₁ and h₂ are filters, x₁ and x₂ are data sources, and w is arandom noise term. For example, for a single carrier communicationssystem, h is the transmit pulse and x constitutes the transmission bittrain. In another system, h may be a filter that models the convolutionof the transmit pulse from an antenna, the communications channel andthe received front-end filters sampled at a rate to obtain a basebandsignal. In another system, h may be a baseband radar pulse such as achirp or a match-filter response, and the data source x is thereflectivity of the radar scene. For the scenario in which a radarsystem and a wireless communications system share a frequency spectrumor two radar systems or two wireless communications system share afrequency spectrum, the filters h₁ and h₂ are meant to overlap infrequency. Objective of example embodiments described herein is todetermine signal sources x₁ and x₂.

At S505, the radar receiver 122 may also retrieve a plurality of systemparameters to be utilized in determining the first and second signals(data sources), as will be described. The plurality of system parametersmay be stored in the memory of the radar receiver 122 (e.g., from thestorage medium device 345 described above with reference to FIG. 3) andthus may be retrieved from the memory. The plurality of systemparameters may include, but is not limited to, the frequency response ofa two linear time-invariant systems (H₁ and H₂).

In one example embodiment and in order to determine the data source x₁and x₂, the radar receiver 122 determines a solution (a pair of vectorsx₁ and x₂) that minimize the following cost function among all possiblevalues of vectors x₁ and x₂.J(x ₁ , x ₂)=θ(y, h ₂ , h ₂ , x ₁ , x ₂)+φ(λ₁ , x ₁, λ₂ , x ₂)   (2)where θ is a data-fidelity term and φ is a regularization function, eachof which may be determined as will be described below. Cost functions(which may also be referred to as optimization cost functions) may beformulated in terms of analysis or synthesis regularization terms or amixture thereof. Example embodiments herein are meant to describe somespecific cases of the formulation of optimization costs functions andthe choice of analysis or synthesis terms or a combination thereofpresented herein, are demonstrative and are not meant to be limiting.Alternatively, an optimization cost function may be formulated from aBayesian estimation theory perspective and estimate the desired signal(e.g. through the maximum a posteriori (MAP) estimate). Such alternativeformulations of the cost function and corresponding solutions may bederived by those skilled in the art and example embodiments of theoptimization cost function formulation presented herein, are meant to bedemonstrative only and thus are not meant to be limiting.

In order to formulate the cost function given in Equation (2), theground receiver 122 determines a data fidelity term at S510. In oneexample embodiment, an energy cost function may be selected for the datafidelity term by the radar receiver 122. Accordingly, Equation (2) maybe rewritten as shown below:

$\begin{matrix}{\left\{ {x_{1},x_{2}} \right\} = {\underset{x_{1},x_{2}}{\arg\;\min}\left\{ {{\frac{1}{2}{{y - {h_{1}*x_{1}} - {h_{2}*x_{2}}}}_{2}^{2}} + {\varphi\left( {\lambda_{1},x_{1},\lambda_{2},x_{2}} \right)}} \right\}}} & (3)\end{matrix}$

Denoting Discrete Fourier Transform (DFT) as F and the inverse thereofas F^(H) having the following property upon normalization,FF^(H)=F^(H)F=I   (4)and the following property of DFT,y(n)=(h*x)(n)⇔Y(k)=√{square root over (N)}X(k)H(k)   (5)using the Parseval property of the unitary DFT, Equation (3) may berewritten as:

$\begin{matrix}{\left\{ {x_{1},x_{2}} \right\} = {\underset{x_{1},x_{2}}{\arg\;\min}\left\{ {{\frac{1}{2}{{Y - {\sqrt{N}X_{1}H_{1}} - {\sqrt{N}X_{2}H_{2}}}}_{2}^{2}} + {\varphi\left( {\lambda_{1},x_{1},\lambda_{2},x_{2}} \right)}} \right\}}} & (6)\end{matrix}$where H_(i) (i=1 and 2) is a diagonal matrix with the elements H_(i)(k)on the diagonal, and X_(i) are the DFT of the signals x_(i). Theregularization parameters λ₁ and λ₂ have positive values, which may bedetermined as will be described below.

At S515, the radar receiver 122 determines a regularization function tobe used in Equation (6). In one example embodiment, the radar receiver122 may determine the regularization term that depends on the statisticsof the under-lying source and allows for separation of sources from theconvolutive mixture. Furthermore, the radar receiver 122 may determinethe regularization function as a combination of regularization functionswith different regularization parameter weights, and the regularizationfunctions may be any one of l₁ norm, nuclear norm, other sparsitypromoting functions including the l₁ norm, non-convex penalties, groupsparse functions, total variation (in range, Doppler, CPI or scan,etc.), mixed norms, Huber loss functions, sparsity in a transform domainsuch as wavelets and Fourier domain, sparsity using prior knowledge suchas clutter maps, structure in time-frequency transforms, etc. Theregularization functions may further be determined depending on thesignals being separated.

Furthermore, in one example embodiment, the l₁ norm may be used as theregularization function, which weights each element by the non-negativeregularization parameter vector λ_(i). This allows for time-varyingregularization parameter as a function of the data index rather than afixed regularization parameter for all time. In using l₁ norm theassumption may be that the signals may be parsimoniously represented bythe filters of the convolutive mixture.

At S520, the radar receiver 122 forms the cost functions of Equation (6)based on the fidelity term determined at S510, the regularizationfunction determined at S515.

The setting of the regularization parameter λ_(i) (for i=1 and 2, forexample) may depend on system parameters such as the noise variance ofthe system, the waveform filters used and the auto-correlation functionof the colored noise. For separable norms such as the l₁ norm, λ_(i) maybe a higher dimensional vector that weights each element differently,thus allowing for the regularization parameter λ_(i) to varyelement-wise, and may further be useful when the noise level or out ofband interference levels change. The value of the regularizationparameter λ_(i) may also be different for different regularizationfunctions. One method of setting the regularization parameter is throughempirical studies that may be used for different scenarios of spectrumoverlap, relative power of the non-overlapping spectrum portions, thewaveform filter, etc.

Another method of setting the regularization parameter λ_(i) is aformula based on system parameters. Another method of setting theregularization parameter λ_(i) is to test several different values ofλ_(i) and ascertain the optimal value of the regularization parameterλ_(i), from among the test values of the regularization parameter λ_(i),and the solutions of the costs function by means of statistical tests.For example, in wireless communications, the statistical test may be thecyclic redundancy check (CRC) and soft/hard error correction codemetrics. In radar, the statistical test may be a function of thecorrelation between the transmit waveform and the estimated radar scene.Other statistical tests (e.g. generalized cross validation, thediscrepancy principle, the L-curve criterion, normalized cumulativeperiodogram), which are known to those skilled in the art, may also beused. Henceforth, the choice of the regularization parameter λ_(i) doesnot change the form of the optimization function and those skilled inthe art may use such methods to set the regularization parameter λ_(i)for different radar and wireless spectrum sharing scenarios.

Thereafter, at S525, the radar receiver 122 determines the data sourcesx₁ and x₂ (first and second signals, from the mixture signal received atS500). In one example embodiment, the radar receiver 122 determines thedata sources x₁ and x₂ as described below using an iterative algorithm.

To minimize the cost function of Equation (6), the radar receiver 122may apply the alternating direction method of multipliers (ADMM). Aniterative algorithm to solve Equation (6) may be derived using ADMM.ADMM is equivalent to or closely related to many other algorithms,including but not limited to, dual decomposition, the method ofmultipliers, Douglas-Rachford splitting, Spingarn's method of partialinverses, Dykstra's alternating projections, Bregman iterativealgorithms for l₁ problems, proximal methods, etc. Accordingly, exampleembodiments are not limited to applying ADMM but may instead usemodified version of the ADMM in order to speed up convergence of theiterative algorithm (e.g., via modifying step-size parameters ρ₁ and ρ₂parameters, which will be described below). Furthermore, other numericaland sparsity optimization methods for minimizing Equation (6), may beused as apparent for those skilled in the art.

Applying the ADMM, the radar receiver 122 may obtain the followingiterative algorithm, in which auxiliary variables d₁ and d₂ are definedand initialized to 0 and Equations (7)-(10) are repeated in an iterativeprocess until a convergence criterion is met.

$\begin{matrix}{\left\{ {x_{1},x_{2}} \right\} = {\underset{x_{1},x_{2}}{\arg\;\min}\;\left\{ {{\frac{1}{2}{{Y - {\sqrt{N}X_{1}H_{1}} - {\sqrt{N}X_{2}H_{2}}}}_{2}^{2}} + {\frac{\rho_{1}}{2}{{x_{1} - u_{1} - d_{1}}}_{2}^{2}} + {\frac{\rho_{2}}{2}{{x_{2} - u_{2} - d_{2}}}_{2}^{2}}} \right\}}} & (7) \\{\left\{ {u_{1},u_{2}} \right\} = {\underset{u_{1},u_{2}}{{\arg\;\min}\mspace{14mu}}\left\{ {{\varphi\;\left( {\lambda_{1},x_{1},\lambda_{2},x_{2}} \right)} + {\frac{\rho_{1}}{2}{{x_{1} - u_{1} - d_{1}}}_{2}^{2}} + {\frac{\rho_{2}}{2}{{x_{2} - u_{2} - d_{2}}}_{2}^{2}}} \right\}}} & (8) \\{\mspace{85mu}{d_{1} = {d_{1} - \left( {x_{1} - u_{1}} \right)}}} & (9) \\{\mspace{85mu}{d_{2} = {d_{2} - \left( {x_{2} - u_{2}} \right)}}} & (10)\end{matrix}$

The convergence criterion may be a configurable variable determinedbased on empirical studies. In one example embodiment, satisfying theconvergence criteria may be achieved by monitoring the change in thevariables of the iterative loop using an appropriate norm.Alternatively, a fixed number of iterations may be used as theconvergence criteria. However, the convergence criteria is not limitedto the examples provided above and may include any other convergencecriteria.

In Equations (7) and (8), ρ₁ and ρ₂ are step-size (which may also bereferred to as multiplier) parameters. The ADMM algorithm will convergefor any step-size parameter ρ_(i). (for i=1,2, for example). However theconvergence rate may differ for different values of the step-sizeparameter ρ_(i). The step-size parameter ρ_(i) may be chosen based onempirical studies or as a function of systems parameters (e.g. noisevariance). Alternatively, the step-size parameter ρ_(i) may be chosenadaptively in each iteration of the ADMM algorithm based on functions ofthe difference between variables in different iterations of the ADMMloop.

In one example embodiment and in order to solve Equation (7), the radarreceiver 122 utilizes the Parseval's property of the DFT. Accordingly,Equation (7) may be written as:

$\begin{matrix}{\left\{ {x_{1},x_{2}} \right\} = {{\underset{x_{1},x_{2}}{\arg\;\min}\left\{ {{\frac{1}{2}{{Y - {\sqrt{N}X_{1}H_{1}} - {\sqrt{N}X_{2}H_{2}}}}_{2}^{2}} + {\frac{\rho_{1}}{2}{{x_{1} - u_{1} - d_{1}}}_{2}^{2}} + {\frac{\rho_{2}}{2}{{x_{2} - u_{2} - d_{2}}}_{2}^{2}}} \right\}} = {\underset{x_{1},x_{2}}{\arg\;\min}\left\{ {{\frac{1}{2}{{Y - {\sqrt{N}X_{1}H_{1}} - {\sqrt{N}X_{2}H_{2}}}}_{2}^{2}} + {\frac{\rho_{1}}{2}{{X_{1} - U_{1} - D_{1}}}_{2}^{2}} + {\frac{\rho_{2}}{2}{{X_{2} - U_{2} - D_{2}}}_{2}^{2}}} \right\}}}} & (11)\end{matrix}$where X_(i), U_(i), D_(i) are the DFTs of x_(i), u_(i) and d_(i).Equation (11) is a least-squares problem that is pairwise separable,which is reducible to a set of independent minimization problems eachhaving two variables as shown in Equation (12):

$\begin{matrix}{{{\frac{1}{2}{{Y - {\sqrt{N}X_{1}H_{1}} - {\sqrt{N}X_{2}H_{2}}}}_{2}^{2}} + {\frac{\rho_{1}}{2}{{X_{1} - U_{1} - D_{1}}}_{2}^{2}} + {\frac{\rho_{2}}{2}{{X_{2} - U_{2} - D_{2}}}_{2}^{2}}} = {\frac{1}{2}{\sum\limits_{k}\;\left( {{{{Y(k)} - {\sqrt{N}{X_{1}(k)}{H_{1}(k)}} - {\sqrt{N}{X_{2}(k)}{H_{2}(k)}}}}_{2}^{2} + {\rho_{1}{{{X_{1}(k)} - {U_{1}(k)} - {D_{1}(k)}}}_{2}^{2}} + {\rho_{2}{{{X_{2}(k)} - {U_{2}(k)} - {D_{2}(k)}}}_{2}^{2}}} \right)}}} & (12)\end{matrix}$

In one example embodiment, the pair {X₁(k), X₂(k)} may be found for eachk by solving a 2×2 system of linear equations. Accordingly, the problemof minimizing the function below whose domain is two complex variablesand whose range is a real value, may be considered:

$\begin{matrix}{{{f\left( {x_{1},x_{2}} \right)} = {{{{Y(k)} - {\overset{\rightarrow}{h}\;\overset{\rightarrow}{x}}}}^{2} + {\left( {\overset{\rightarrow}{x} - \overset{\rightarrow}{b}} \right)^{H}{M\left( {\overset{\rightarrow}{x} - \overset{\rightarrow}{b}} \right)}^{H}}}}{where}} & (13) \\{{\overset{\rightarrow}{x} = \begin{bmatrix}{X_{1}(k)} \\{X_{2}(k)}\end{bmatrix}},\mspace{14mu}{\overset{\rightarrow}{h} = \begin{bmatrix}{\sqrt{N}{H_{1}(k)}} & {\sqrt{N}{H_{2}(k)}}\end{bmatrix}},{\overset{\rightarrow}{b} = \begin{bmatrix}{{U_{1}(k)} + {D_{1}(k)}} \\{{U_{2}(k)} + {D_{2}(k)}}\end{bmatrix}},\mspace{14mu}{M = \begin{bmatrix}\rho_{1} & 0 \\0 & \rho_{2}\end{bmatrix}}} & (14)\end{matrix}$

Setting the gradient of f to zero, an expression for the minimizer x maybe obtained as:{right arrow over (x)}=({right arrow over (h)} ^(H) {right arrow over(h)}+M)⁻¹(Y(k){right arrow over (h)} ^(H) +M{right arrow over (b)})  (15)

Using the Matrix Inverse lemma and simplifying Equation (15), thefollowing solution for the minimization of Equation (11) may beobtained:

$\begin{matrix}{\begin{bmatrix}{X_{1}(k)} \\{X_{2}(k)}\end{bmatrix} = {\begin{bmatrix}{R_{1}(k)} \\{R_{2}(k)}\end{bmatrix} - {\begin{bmatrix}{{H_{1}(k)}*\left( \rho_{1}^{- 1} \right)} \\{{H_{2}(k)}*\left( \rho_{2}^{- 1} \right)}\end{bmatrix}{G(k)}\left( {{{H_{1}(k)}{R_{1}(k)}} + {{H_{2}(k)}{R_{2}(k)}}} \right)}}} & (16)\end{matrix}$where * denotes complex conjugate,

$\begin{matrix}{{R_{i}(k)} = {{\frac{\sqrt{N}}{\rho_{i}}{H_{i}(k)}*{Y(K)}} + {U_{i}(k)} + {D_{i}(k)}}} & (17) \\{and} & \; \\{{G(k)} = \left( {\frac{1}{N} + \frac{{{H_{1}(k)}}^{2}}{\rho_{1}} + \frac{{{H_{2}(k)}}^{2}}{\rho_{2}}} \right)^{- 1}} & (18)\end{matrix}$

Furthermore, variable T(k) may be defined as:T(k)=G(k)(H ₁(k)R ₁(k)+H ₂(k)R ₂(k))   (19)

Using Equations (17)-(19):

$\begin{matrix}{{{X_{i}(k)} = {{{R_{i}(k)} - {\frac{1}{\rho_{i}}{H_{i}(k)}*{T(k)}\mspace{14mu}{for}\mspace{14mu} i}} = 1}},2} & (20)\end{matrix}$

Furthermore, given a separable regularization function determined atS515, Equation (8) may be written as:

$\begin{matrix}{\left\{ {u_{1},u_{2}} \right\} = {\underset{u_{1},u_{2}}{{\arg\;\min}\;}\left\{ {{\varphi_{1}\left( {\lambda_{1} \odot u_{1}} \right)} + {\varphi_{2}\left( {\lambda_{2} \odot u_{2}} \right)} + {\frac{\rho_{1}}{2}{{x_{1} - u_{1} - d_{1}}}_{2}^{2}} + {\frac{\rho_{2}}{2}{{x_{2} - u_{2} - d_{2}}}_{2}^{2}}} \right\}}} & (21)\end{matrix}$

The decoupled Equation (21) may be written as:

$\begin{matrix}{\left\{ u_{i} \right\} = {\underset{u_{i}}{\arg\;\min}\left\{ {{\varphi_{i}\left( {\lambda_{i} \odot u_{i}} \right)} + {\frac{\rho_{i}}{2}{{x_{i} - u_{i} - d_{i}}}_{2}^{2}}} \right\}}} & (22)\end{matrix}$for i=1, 2.

Using the concept of the proximity operator, Equation (22) may bere-written in the notation as:

$\begin{matrix}{{{prox}_{\varphi,\lambda,\rho}(z)} = {\underset{u}{\arg\;\min}\left\{ {{\varphi\left( {\lambda \odot u} \right)} + {\frac{\rho}{2}{{z - u}}_{2}^{2}}} \right\}}} & (23)\end{matrix}$

Closed-form expressions of the proximity operators of various functionsexist and if closed-form expressions are not derivable, numericaloptimization methods to obtain an estimate of the proximity operator maybe utilized.

Using l₁ norm as the regularization function, Equation (22) may bewritten as:

$\begin{matrix}{\left\{ u_{i} \right\} = {\underset{u_{i}}{\arg\;\min}\left\{ {{{\lambda_{i} \odot u_{i}}}_{1} + {\frac{\rho_{i}}{2}{{x_{i} - u_{i} - d_{i}}}_{2}^{2}}} \right\}}} & (24)\end{matrix}$where ⊚ denotes point-wise multiplication, the solution of which isgiven explicitly in terms of the soft-threshold rule:u _(i)=soft(x _(i) −d _(i), λ_(i)/ρ_(i))   (25)

The soft-threshold rule may be applied element-wise to vectors, matricesand higher dimensional vectors. Defining “./” as point-wise division,“.^” as point-wise power for a vector and defining v_(i) asv_(i)=u_(i)+d_(i), the radar receiver 122, using the Equations describedabove, may determine the data source x₁ and x₂ by applying the followingiterative process, which may also be denoted to as “dual-deconvolutionwith denoising” process.

The iterative algorithm defines several auxiliary variables. First avariable G is defined as shown in Equation (26),

$\begin{matrix}{G = {1 \cdot {/\left( {\frac{1}{N} + \;\frac{{H_{1}} \cdot^{\bigwedge 2}}{\rho_{1}} + \frac{{H_{2}} \cdot^{\bigwedge 2}}{\rho_{2}}} \right)}}} & (26)\end{matrix}$and variables x_(i) ⁽⁰⁾=x_(i) ^(init), d_(i) ⁽⁰⁾=d_(i) ^(init) areinitialized for i=1,2. The radar receiver 122, may further define avariable k indicative of the number of iterations of the iterativeprocess (i.e., k is a counter) and initialize k to 0. Thereafter anduntil a convergence criterion, as described above, is satisfied, theradar receiver 122 iteratively solves Equations (27)-(32), as shownbelow, with k being incremented by 1 after each iteration.

$\begin{matrix}{k = {k + 1}} & (27) \\{{v_{i}^{(k)} = {{{{soft}\mspace{11mu}\left( {{x_{i}^{({k - 1})} - d_{i}^{({k - 1})}},\frac{\lambda_{i}}{\rho_{i}}} \right)} + {d_{i}^{({k - 1})}\mspace{14mu}{for}\mspace{14mu} i}} = 1}},2} & (28) \\{{R_{i}^{(k)} = {{{Fv}_{i}^{(k)} + {\left( {\sqrt{N}/\rho_{i}} \right){H_{i}^{*} \odot Y}\mspace{14mu}{for}\mspace{14mu} i}} = 1}},2} & (29) \\{T^{(k)} = {G \odot \left( {{H_{1} \odot R_{1}^{(k)}} + {H_{2} \odot R_{2}^{(k)}}} \right)}} & (30) \\{{x_{i}^{(k)} = {{{F^{H}\left\lbrack {R_{i}^{(k)} - {\left( {1/\rho_{i}} \right){H_{i}^{*} \odot T^{(k)}}}} \right\rbrack}\mspace{14mu}{for}\mspace{14mu} i} = 1}},2} & (31) \\{{d_{i}^{(k)} = {{v_{i}^{(k)} - {x_{i}^{(k)}\mspace{14mu}{for}\mspace{14mu} i}} = 1}},2} & (32)\end{matrix}$

x_(i) ^((k)) for i=1 and 2 at step k where the convergence criterion ismet, represent values of data sources x₁ and x₂ that minimize the costfunction given in Equation (6).

The “dual-deconvolution with denoising” process described above withreference to Equations (26)-(32) may easily be extended to anM-deconvolution of M datasources from a convolutive mixture. Moregenerally, if the convolute mixture involves M sources, Equations (26)and Equations (30) may be modified as follows:

$\begin{matrix}{G = {1 \cdot {/\left( {\frac{1}{N} + {\sum\limits_{i = 1}^{M}\;\frac{{H_{i}} \cdot^{\bigwedge 2}}{\rho_{i}}}} \right)}}} & (33) \\{T^{(k)} = {G \odot \left( {\sum\limits_{i = 1}^{M}{H_{i} \odot R_{i}^{(k)}}} \right)}} & (34)\end{matrix}$With i=1, . . . , M for all indexed variables in Equations (26)-(32).

Accordingly and upon determining data sources x₁ and x₂, as describedabove, at S530, the radar receiver 122 may process informationassociated with one or more of the data sources x₁ and x₂,determined/estimated at S525. For example, the radar receiver 122 mayanalyze the determined radar signal to detect objects corresponding tothe underlying purpose of the radar system, track/monitor variablesand/or objects of interest (e.g., speed of cars, airplanes, ships,etc.). However, the processing of the radar signal is not limited to theexamples described above but may encompass any appropriate type ofanalysis of the determined/estimated radar signal in order toextract/study/monitor information included in or associated with thedetermined/estimated radar signal. Furthermore, when the receiver is areceiver of a non-radar system (e.g., a wireless communications system),the receiver, upon determining the data sources, may process anyinformation associated with the determined data sources (e.g.,transmission of voice data, communication between network components ina wireless communications system, etc.).

In one example embodiment, the mixture signal received at the receiverradar 122 may be in the form of modulated waveforms, as shown below:

$\begin{matrix}{{{y(n)} = {{\left( {h_{1}*x_{1}} \right)(n)e^{j\;\omega_{1}n}} + {\left( {h_{2}*x_{2}} \right)(n)e^{j\;\omega_{2}n}} + {w(n)}}},{{{for}\mspace{14mu} n} = 0},\ldots\mspace{14mu},{N - 1}} & (35)\end{matrix}$where ω₁ and ω₂ are modulating waveforms. In the discrete-time Fouriertransform (DTFT) domain, Equation (35) may be written as:Y(ω)=H ₁(ω−ω₁)X ₁(ω−ω₁)+H ₂(ω−ω₂)X ₂(ω−ω₂)+W(ω)   (36)

Consequently, the cost function of Equation (3) may be reformulated as:

$\begin{matrix}{{C\left( {x_{1},x_{2}} \right)} = {\underset{\overset{\_}{x_{1}},\overset{\_}{x_{2}}}{\arg\;\min}\left\{ {{\frac{1}{2}{{y - {\overset{\_}{h_{1}}*\overset{\_}{x_{1}}} + {\overset{\_}{h_{2}}*\overset{\_}{x_{2}}}}}_{2}^{2}} + {\varphi\left( {\lambda_{1},{\overset{\_}{x}}_{1},\lambda_{2},{\overset{\_}{x}}_{2}} \right)}} \right\}}} & (37)\end{matrix}$where h _(i) and x _(i) are the modulated forms of h _(i) and x _(i),given by:h _(i)(n)=h _(i)(n)e ^(jω) ^(i) ^(n) , x _(i)(n)=x _(i)(n)e ^(jω) ^(i)^(n)   (38)

Based on these definitions, Equation (6) may be rewritten as:

$\begin{matrix}{\left\{ {\overset{\_}{x_{1}},\overset{\_}{x_{2}}} \right\} = {\underset{\overset{\_}{x_{1}},\overset{\_}{x_{2}}}{\arg\;\min}\left\{ {{\frac{1}{2}{{Y - {\sqrt{N}{\overset{\_}{X}}_{1}{\overset{\_}{H}}_{1}} - {\sqrt{N}{\overset{\_}{X}}_{2}{\overset{\_}{H}}_{2}}}}_{2}^{2}} + {\varphi\left( {\lambda_{1},{\overset{\_}{x}}_{1},\lambda_{2},{\overset{\_}{x}}_{2}} \right)}} \right\}}} & (39)\end{matrix}$where X _(i) and H _(i) are the DFT of x _(i) and h _(i), respectively.Since Equation (39) is essentially the same as the cost function ofEquation (6), the “dual-deconvolution with denoising” process maysimilarly be applied to find the data sources x₁ , x₂ as solutions toEquation (39). x ₁ and x ₂ may then be demodulated using known methodsin order to obtain the data-source signals.

In one example embodiment, the noise w(n) in Equation (1) is consideredto be colored noise with a power spectral density (PSD) P_(w). In orderto account for w(n), the radar receiver 122 may weigh the frequencies ofthe data fidelity term by a function of P_(w). In one exampleembodiment, the radar receiver 122 may weigh the frequencies of the datafidelity term by the reciprocal of the square root of P_(w) while othermethods of weighting the frequencies of the data fidelity term based onthe power spectral density of the total noise (consisting of systemnoise and noise induced by co-existing/co-located systems) may also beused.

Using the reciprocal of P_(w), Equation (6) may then be modified asfollows:

$\begin{matrix}{\left\{ {x_{1},x_{2}} \right\} = {\underset{x_{1},x_{2}}{\arg\;\min}\mspace{11mu}\left\{ {{\frac{1}{2}{{\frac{1}{\sqrt{P_{w}}}\left( {Y - {\sqrt{N}X_{1}H_{1}} - {\sqrt{N}X_{2}H_{2}}} \right)}}_{2}^{2}} + {\varphi\left( {\lambda_{1},x_{1},\lambda_{2},x_{2}} \right)}} \right\}}} & (40)\end{matrix}$

Here, the term √{square root over (P_(w))} may be absorbed in Y, H₁ andH₂, and the same “dual-deconvolution with denoising” process, asdescribed above, may be applied to determine the data sources x₁ and x₂.

FIG. 6 describes a method of determining data sources from a mixturesignal, according to an example embodiment.

In FIG. 6, S600 and S605 is the same as S500 and S505. Therefore, forthe sake of brevity, S600 and S605 will not be further described.

In the cost function described herein with reference to FIG. 6, aperfect reconstruction constraint instead of the data fidelity term atS510 of FIG. 5 may be used. The perfect reconstruction constraint may besuitable for separation of signals from one another in a receivedmixture signal when there is no noise or small amounts of noise present(i.e., w(n) in Equation (1) is small and negligible). Because the noiseterm w(n) is negligible, the method of FIG. 6 does not include a stepsimilar to S510 for determining a data fidelity term.

Accordingly, at S610, the radar receiver 122 may determine aregularization term. In one example embodiment, the regularization termmay be determined as the l₁ norm where φ(λ₁, x₁, λ₂, x₂)=∥λ₁ ⊚ x₁∥₁+∥λ₂⊚ x₂∥₁.

Thereafter, at S615, the radar receiver 122 forms the cost functionbased on the regularization term determined at S610. In one exampleembodiment, the radar receiver may form the cost function as shown inEquation (41) below.

$\begin{matrix}{{\left\{ {x_{1},x_{2}} \right\} = {\underset{x_{1},x_{2}}{\arg\;\min}\mspace{11mu}\left\{ {\varphi\left( {\lambda_{1},x_{1},\lambda_{2},x_{2}} \right)} \right\}}}{{s.t.\mspace{11mu} y} = {{h_{1}*x_{1}} + {h_{2}*x_{2}}}}{{\lambda_{1} + \lambda_{2}} = {{1\mspace{14mu}{and}\mspace{14mu}\lambda_{i}} \geq 0}}} & (41)\end{matrix}$

At S620, the radar receiver 122 determines data sources x₁ and x₂ (firstand second signals). In one example embodiment, the radar receiver 122determines the data sources x₁ and x₂, by finding a solution thatminimizes Equation (41) using an iterative process similar to the“dual-deconvolution with denoising” process described above withreference to FIG. 5. This iterative process may be denoted as the“dual-deconvolution with perfect reconstruction” process. An example ofthe “dual-deconvolution with perfect reconstruction” process fordetermining the data sources x₁ and x₂, according to the method of FIG.6 for the l₁ norm regularization function is described below.

According to the “dual-deconvolution with perfect reconstruction”process, a variable G is defined as given in Equation (42). Thereafter,a plurality of variables x_(i) ⁽⁰⁾ and d_(i) ⁽⁰⁾ are initialized fori=1,2. The radar receiver 122, may further define a variable kindicative of the number of iterations of the iterative process (i.e., kis a counter) and initialize k to 0. Thereafter, until a convergencecriterion (which may be as defined above) is met, the process providedby Equations (43)-(48) is repeated with k being incremented by 1 aftereach iteration.

$\begin{matrix}{G = \left( {\frac{{H_{1}} \cdot^{\bigwedge 2}}{\rho_{1}} + \frac{{H_{2}} \cdot^{\bigwedge 2}}{\rho_{2}}} \right)} & (42) \\{k = {k + 1}} & (43) \\{{v_{i}^{(k)} = {{{{soft}\left( {{x_{i}^{({k - 1})} - d_{i}^{({k - 1})}},\frac{\lambda_{i}}{\rho_{i}}} \right)} + {d_{i}^{({k - 1})}\mspace{14mu}{for}\mspace{14mu} i}} = 1}},2} & (44) \\{{V_{i}^{(k)} = {{F\; v_{i}^{(k)}\mspace{14mu}{for}\mspace{14mu} i} = 1}},2} & (45) \\{T^{(k)} = {\left( {\frac{Y}{\sqrt{N}} - \left\lbrack {{H_{1} \odot V_{1}^{(k)}} + {H_{2} \odot V_{2}^{(k)}}} \right\rbrack} \right) \cdot {/G}}} & (46) \\{{x_{i}^{(k)} = {{{F^{H}\left\lbrack {{\left( {1/\rho_{i}} \right){H_{i}^{*} \odot T^{(k)}}} + V_{i}^{(k)}} \right\rbrack}\mspace{14mu}{for}\mspace{14mu} i} = 1}},2} & (47) \\{{d_{i}^{(k)} = {{v_{i}^{(k)} - {x_{i}^{(k)}\mspace{14mu}{for}\mspace{14mu} i}} = 1}},2} & (48)\end{matrix}$

x_(i) ^((k)) for i=1 and 2 at step k where the convergence criterion ismet, represent values of data sources x₁ and x₂ that minimize the costfunction given by Equation (41).

Thereafter, at S625, the radar receiver 122 may perform the samefunction as described above with reference to S530 in FIG. 5. Therefore,for the sake of brevity, S625 will be not be described any further.

FIG. 7 describes a method of determining data sources from a mixturesignal, according to an example embodiment.

With respect to FIG. 7 and as will be described below, the mixturesignal received at a receiver is a combination of at least oneconvolutive process and at least one signal that is effectivelyrepresented and/or compressible by a transform (basis, frame,dictionary), where the transform may be undercomplete, complete orovercomplete. For illustrative purposes, the special case of applying anovercomplete transform will be described.

An objective herein with reference to FIG. 7 is to obtain data-sourcecoefficients x₁ and a set of transform coefficients c₂. For illustrativepurposes, FIG. 7 will be described with reference to two signals (onebeing a convolutive process and one being a set of transformcoefficients). However, inventive concepts may be easily extended to acase of M=M₁+M₂ signals where M₁ components are modeled as convolutiveprocesses and M₂ components are modeled as arising from transforms. Withrespect to the transforms, the identity matrix and orthogonal transformsare encompassed by the overcomplete transform discussed.

At S700, the radar receiver 122 receives a mixture signal y that is acombination of a convolution of a data filter h with a data source x₁plus a signal that is represented by an overcomplete Parseval's frame Swith transform coefficients c₂ plus a noise term w, as shown below.y(n)=(h*x ₁)(n)+x ₂(n)+w(n)=(h*x ₁)(n)+(Sc ₂)_(n) +w(n)   (49)where S satisfies:SS^(H)=αI, α>0   (50)and Sc₂=x₂.

The frame S may be a fast transform such as the overcomplete InverseFast Fourier Transform (IFFT). In general, it may be advantageous forreal-time systems that S and S^(H) be computed using fast transformssuch as Fast Fourier Transform (FFT), Discrete Cosine Transform (DCT),wavelet transform, etc.

At S705, the radar receiver 122 may retrieve further inputs including aplurality of system parameters. The plurality of system parameters maybe stored in the memory of the radar receiver 122 (e.g., from thestorage medium device 345 described above with reference to FIG. 3) andthus may be retrieved from the memory. The plurality of systemparameters may include, but is not limited to, the frequency response ofa linear time-invariant system (H) corresponding to h and the transform(S).

At S710, the radar receiver 122 may form a cost function fordetermining/extracting the data sources from the mixture signal receivedat S700. In one example embodiment, the cost function to be formed inorder to find the two data sources {x₁, x₂=Sc₂}, is given by Equation(51), which is in the frequency domain:

$\begin{matrix}{\left\{ {x_{1},c_{2}} \right\} = {{\underset{x_{1},c_{2}}{\arg\;\min}\frac{1}{2}{{Y - {\sqrt{N}{{diag}(H)}F\; x_{1}} - {FSc}_{2}}}_{2}^{2}} + \left( {\lambda_{1},x_{1},\lambda_{2},c_{2}} \right)}} & (51)\end{matrix}$

where the data fidelity term ½∥Y−√{square root over(N)}diag(H)Fx₁−FSc₂∥₂ ² and φ(λ₁, x₁, λ₂, c₂) may de determined in asimilar manner as described above with reference to FIG. 5. It is notedthat with suitable zero-padding, the linear convolution given byEquation (49) is transformed to a circular convolution that is performedas a multiplication in the Fourier domain as in Equation (51).

At S715, the radar receiver 122 may determine the data sources {x₁,x₂=Sc₂}, (first and second signals) by finding a solution to Equation(51). In one example embodiment, the radar receiver 122 may find thesolution to Equation (51) (i.e., find data sources x₁ and x₂) byapplying an iterative process to find the solution to equation (51). Thesolution to Equation (51) may be a pair of vectors (x₁ and c₂) thatminimize the cost function given by Equation (51) over all possiblevalues of x₁ and c₂.

The solution to Equation (51) may be obtained via an iterative processreferred to as the “basis pursuit denoising-deconvolution (BPD-Deconv)”process. In the BPD-Deconv process, discrete Fourier Transform (DFT) isdenoted as F, the inverse thereof is denoted as F^(H) and diag isdenoted as the diagonal operator.

Herein, the size of S is considered to be N×L where L≥N and N is thelength of the mixture signal y. Variables A, A₁ and A₂, are defined byEquation (52)A=[A ₁ A ₂] where A ₁ =√{square root over (N)}diag(H)F and A ₂=FS   (52)where the operators embedded in A₁ and A₂ may be performed using fasttransforms. In one example embodiment, the regularization function usedin Equation (51) may be a separable l₁ norm. The following property alsoholds:

$\begin{matrix}{{AA}^{H} = {{{\sqrt{N}{{diag}\left( {{H} \cdot^{\bigwedge 2}} \right)}} + {{FSS}^{H}F}} = {{{\sqrt{N}{{diag}\left( {{H} \cdot^{\bigwedge 2}} \right)}} + {\alpha\; I}} = {{diag}\left( {N{{H} \cdot^{\bigwedge 2}{+ \alpha}}} \right)}}}} & (53)\end{matrix}$

Using ADMM with similar steps to the “dual-deconvolution with denoising”process described above with reference to FIG. 5, the matrix inverselemma and the identity above given by Equation (53), the radar receiver122 may apply the following iterative process for separating the mixturesignal and determining x₁ and x₂.

The iterative process based on the BPD-Deconv process includes definingG, as given by Equation (54) and variables y₁ and y₂ as given byEquations (55) and (56).

$\begin{matrix}{G = {1 \cdot {/\left( {1 + \frac{N{{H_{1}} \cdot^{\bigwedge 2}}}{\rho_{1}} + \frac{\alpha}{\rho_{2}}} \right)}}} & (54) \\{y_{1} = {A_{1}^{H}{Fy}}} & (55) \\{y_{2} = {A_{2}^{H}{Fy}}} & (56)\end{matrix}$

The iterative process further includes the initializing variable d fori=1,2 as d_(i) ⁽⁰⁾=d_(i) ^(init). The variables x₁ ⁽⁰⁾=x₁ ^(init) and c₂⁽⁰⁾=c₂ ^(init) are also initialized. The radar receiver 122, may furtherdefine a variable k indicative of the number of iterations of theiterative process (i.e., k is a counter) and initialize k to 0.Thereafter and until a convergence criterion(which may be the same asdescribed above), the radar receiver 122 repeats Equations (57)-(65) ineach step, with k being incremented by 1 after completion of eachiteration of the iterative process.

$\begin{matrix}{k = {k + 1}} & (57) \\{u_{1}^{(k)} = {{soft}\left( {{x_{1}^{({k - 1})} + d_{1}^{({k - 1})}},\frac{\lambda_{1}}{\rho_{1}}} \right)}} & (58) \\{u_{2}^{(k)} = {{soft}\left( {{c_{2}^{({k - 1})} + d_{2}^{({k - 1})}},\frac{\lambda_{2}}{\rho_{2}}} \right)}} & (59) \\{{b_{i}^{(k)} = {{y_{i} + {{\rho_{i}\left( {u_{i}^{({k - 1})} - d_{i}^{({k - 1})}} \right)}\mspace{14mu}{for}\mspace{14mu} i}} = 1}},2} & (60) \\{Z^{(k)} = {G \odot \left( {{A_{1}{b_{1}^{(k)}/\rho_{1}}} + {A_{2}{b_{2}^{(k)}/\rho_{2}}}} \right)}} & (61) \\{x_{1}^{(k)} = {\left( {b_{1}^{(k)} - {A_{1}^{H}Z^{(k)}}} \right)/\rho_{1}}} & (62) \\{c_{2}^{(k)} = {\left( {b_{2}^{(k)} - {A_{2}^{H}Z^{(k)}}} \right)/\rho_{2}}} & (63) \\{d_{1}^{(k)} = {d_{1}^{({k - 1})} - u_{1}^{(k)} + x_{1}^{(k)}}} & (64) \\{d_{2}^{(k)} = {d_{2}^{({k - 1})} - u_{2}^{(k)} + c_{2}^{(k)}}} & (65)\end{matrix}$

Thereafter, x₁ ^((k)) and c₂ ^((k)) at step k where the convergencecriterion is met, represent values of data sources x₁ and x₂ (Sc₂=x₂)that minimize the cost function given in Equation (51).

At S720, the radar receiver 122 may perform the same functions asdescribed above with reference to S530 in FIG. 5. Therefore, for thesake of brevity, S720 will not be further described.

FIG. 8 describes a method of determining data sources from a mixturesignal, according to an example embodiment.

In FIG. 8, S800 and S805 is the same as S700 and S705. Therefore, forthe sake of brevity, S800 and S805 will not be further described.

At S810, the radar receiver 122 may form the cost function using perfectreconstruction metric. Accordingly, instead of the cost function givenby Equation (51), the radar receiver 122 forms the following costfunction:

$\begin{matrix}{{\left\{ {x_{1},x_{2}} \right\} = {\underset{x_{1},x_{2}}{\arg\;\min}\mspace{11mu}\left\{ {\varphi\left( {\lambda_{1},x_{1},\lambda_{2},c_{2}} \right)} \right\}}}{{s.t.\mspace{11mu} y} = {{{h_{1}*x_{1}} + {{Sc}_{2}\mspace{14mu}{and}\mspace{14mu}\lambda_{i}}} \geq 0}}} & (66)\end{matrix}$where, as an example embodiment, for the l₁ norm regularizationfunction, the radar receiver 122 may determine the regularization term φas φ(λ₁, x₁, λ₂, c₂)=∥λ₁ ⊚ x₁∥₁+∥λ₂ ⊚ c₂∥₁, as described above withreference to FIG. 6.

The application of perfect reconstruction may be suitable for separationof signals from one another in a received mixture signal when there isno noise or small amounts of noise present (i.e., w(n) in Equation (1)is small and negligible). Accordingly, Equation (66) may not contain adata fidelity term compared to the equation (51) described above withreference to FIG. 7.

At S815, the radar receiver 122 determines data sources x₁ and x₂ (firstand second signals). In one example embodiment, the radar receiver 122determines the data sources x₁ and x₂, by finding a solution thatminimizes Equation (66) using an iterative process similar to theBPD-Deconv process described above with reference to FIG. 7, which maybe referred to as “BPD-Deconv with perfect reconstruction” process. Anexample of the “BPD-Deconv with perfect reconstruction” process fordetermining the data sources x₁ and x₂ for the l₁ norm regularizationfunction according to the method of FIG. 8 is described below.

According to the “BPD-Deconv with perfect reconstruction” process, avariable G is defined, as given by Equation (67). Furthermore, variableY is defined in Equation (68) as the Fourier transform of y.

$\begin{matrix}{G = {1 \cdot {/\left( {\frac{N{{H_{1}} \cdot^{\bigwedge 2}}}{\rho_{1}} + \frac{\alpha}{\rho_{2}}} \right)}}} & (67) \\{Y = {F\; y}} & (68)\end{matrix}$

The “BPD-Deconv with perfect reconstruction” process further includesthe initializing variable d for i=1,2 as d_(i) ⁽⁰⁾=d_(i) ^(init). Thevariables x₁ ⁽⁰⁾=x₁ ^(init) and c₂ ⁽⁰⁾=c₂ ^(init) are also initialized.The radar receiver 122, may further define a variable k indicative ofthe number of iterations of the iterative process (i.e., k is a counter)and initialize k to 0. Thereafter and until a convergencecriterion(which may be the same as described above), the radar receiver122 repeats Equations (69)-(78) in each step, with k being incrementedby 1 after completion of each iteration of the iterative process.

$\begin{matrix}{k = {k + 1}} & (69) \\{u_{1}^{(k)} = {{soft}\mspace{11mu}\left( {{x_{1}^{({k - 1})} + d_{1}^{({k - 1})}},\frac{\lambda_{1}}{\rho_{1}}} \right)}} & (70) \\{u_{2}^{(k)} = {{soft}\mspace{11mu}\left( {{c_{2}^{({k - 1})} + d_{2}^{({k - 1})}},\frac{\lambda_{2}}{\rho_{2}}} \right)}} & (71) \\{{b_{1}^{(k)} = \left( {u_{1}^{(k)} - d_{1}^{({k - 1})}} \right)}\mspace{11mu}} & (72) \\{b_{2}^{(k)} = \left( {u_{2}^{(k)} - d_{2}^{({k - 1})}} \right)} & (73) \\{T^{(k)} = {G \odot \left( {Y - {A_{1}b_{1}^{(k)}} - {A_{2}b_{2}^{(k)}}} \right)}} & (74) \\{x_{1}^{(k)} = {\frac{A_{1}^{H}T^{(k)}}{\rho_{1}} + b_{1}}} & (75) \\{c_{2}^{(k)} = {\frac{A_{2}^{H}T^{(k)}}{\rho_{2}} + b_{2}}} & (76) \\{d_{1}^{(k)} = {d_{1}^{({k - 1})} - u_{1}^{(k)} + x_{1}^{(k)}}} & (77) \\{d_{2}^{(k)} = {d_{2}^{({k - 1})} - u_{2}^{(k)} + c_{2}^{(k)}}} & (78)\end{matrix}$

In one example embodiment, x₁ ^((k)) and c₂ ^((k)) at step k where theconvergence criterion is met, represent values of data sources x₁ and x₂(Sc₂=x₂) that minimize the cost function given in Equation (66).

Thereafter, at S820, the radar receiver 122 may perform the samefunction as described above with reference to S530 in FIG. 5. Therefore,for the sake of brevity, S820 will be not be described any further.

While FIGS. 5-8 are described above with reference to the radar receiver122, example embodiments are not limited thereto. For example themethods described above with reference to FIGS. 5-8, may be implementedat a receiver in any one of the system components of a wirelesscommunications system such as the second system 130 shown in FIG. 1(e.g., a receiver of any one of the UEs 134, the receiver of thewireless access node 132, etc.).

In some example embodiments, there may be more than one system of aparticular technology. For example, in the setting shown in FIG. 1,there may be more than one radar system such as the system 120. In otherwords, there may be two radar systems 130 and the wirelesscommunications system 120 whose signals may be simultaneously andoverlappingly transmitted. Accordingly, a radar receiver 122 of any ofthe radar systems 130 may suppress the radar signals of the other one ofthe radar systems 130 (i.e., undesired radar signal) when implementingexample embodiments for determining/estimating the corresponding radarsignal (i.e., the desired radar signal). In this context, suppressing ofa radar signal may be understood to include eliminating the influence ofthe undesired radar signal sufficiently so that the undesired radarsignal induces minimal detrimental effect on determining/estimating thedesired radar signal.

In one example embodiment, any of the radar systems 120 may suppress theundesired radar signals of the other radar system(s) 120 by adjusting inthe cost function the power spectral densities on frequencies on whichthe undesired radar signals of the other radar system(s) 120 aretransmitted.

In one example embodiment, there may be more than one wirelesscommunications system and a radar system. Accordingly, a receiver at acomponent of any of the wireless communications systems may suppress thesignals associated with the other wireless communications system(s)(i.e., the undesired wireless communications signals), when determiningthe radar signal and subsequently the intended wireless communicationssignal.

In one example embodiment, the receiver at a component of any of thewireless communications systems may suppress the undesired wirelesscommunications signals in a similar manner as described above withreference to the radar systems (e.g., in the cost function adjustingpower spectral densities on frequencies on which the undesired wirelesscommunications signals are transmitted).

In another example embodiment and when the undesired wirelesscommunications signals are sparse, the receiver at a component of any ofthe wireless communications systems may suppress the undesired wirelesscommunications signals by subtracting the sparse undesired wirelesscommunications signals from the intended (desired wirelesscommunications) signal.

Example embodiments described above provide numerous advantages overexisting methods in the art, as described in the Background Section.Some of the advantages are described below.

The example advantages are described with respect to one or more ofexample embodiments described herein. However, example advantages arenot meant to limit all example embodiments described herein. One or moreexample embodiments may provide advantages other than the exampleadvantages described below.

One example advantage provided by example embodiments of inventiveconcepts is enabling separation of sources of a convolutive mixturesusing one observed time-series/channel of observation. In terms of theRF spectrum environment, inventive concepts enable frequency spectrumsharing applications using only one antenna (although exampleembodiments are not limited to one antenna) and multiple uncoordinatedsources transmitting concurrently over the same frequency spectral band.

Another example advantage provided by example embodiments of inventiveconcepts, relative to DSA and DFS technology, is enabling both types ofsystems to operate simultaneously. That is, the signals overlapping intime while overlapping partially or fully in frequency may be recovered.

Furthermore, while example embodiments have been described with a focuson radar and wireless communications systems, inventive concepts applyto other fields and sensors such as biomedical, optics, IR, acoustics,seismic, sonar, ultrasound, speech signals where separation ofcomponents/signals are used for signal processing tasks.

Variations of the example embodiments are not to be regarded as adeparture from the spirit and scope of the example embodiments, and allsuch variations as would be apparent to one skilled in the art areintended to be included within the scope of this disclosure.

What is claimed:
 1. A method comprising: receiving a mixture signal;determining a cost function associated with the mixture signal, anddetermining a first signal and a second signal from the mixture signalbased on the cost function, the first signal being a signal of a firstwireless communication system and the second signal being a signal ofone of a second wireless communication system or a radar system, thefirst and second signals being overlappingly transmitted signals, atleast one of the first signal and the second signal being used forprocessing of information associated with the at least one of the firstsignal and the second signal.
 2. The method of claim 1, wherein theoverlapping transmission of the first signal and the second signalincludes at least one of: a spatial overlap of the first signal and thesecond signal; overlapping transmission of the first signal and thesecond signal in a time domain; and overlapping transmission of thefirst signal and the second signal in a frequency domain.
 3. The methodof claim 1, wherein the first wireless communication system and thesecond wireless system are based on different wireless communicationsstandards.
 4. The method of claim 1, wherein the mixture signal is aconvolutive mixture of a first convolutive process, a second convolutiveprocess and a noise signal, the first convolutive process corresponds toa convolution of the first signal and a first impulse response of thefirst wireless communication system, and the second convolutive processcorresponds to a convolution of the second signal and a second impulseresponse of one of the second wireless communication system or the radarsystem.
 5. The method of claim 4, further comprising: determining thecost function as a function of the first signal and the second signal,and minimizing the cost function, and determining the first signal andthe second signal from among possible sets of values of the first signaland the second signal that minimize the cost function.
 6. The method ofclaim 5, wherein the cost function is based on the received mixturesignal, the first impulse response, the second impulse response, a firstregularization term corresponding to the first signal and a secondregularization term corresponding to the second signal.
 7. The method ofclaim 1, wherein the mixture signal is a combination of a convolutiveprocess, a signal represented by a combination of the second signal anda transform, and a noise signal, the convolutive process corresponds toa convolution of the first signal and an impulse response of the firstwireless communication system, and the transform is at least one of anundercomplete, complete and overcomplete transform.
 8. The method ofclaim 7, further comprising: determining the cost function as a functionof the first signal and the second signal, and determining the firstsignal and the second signal from among possible sets of values of thefirst signal and the second signal that minimize the cost function. 9.The method of claim 8, wherein the cost function is based on thereceived mixture signal, the impulse response, the transform, a firstregularization term corresponding to the first signal and a secondregularization term corresponding to the second signal.
 10. The methodof claim 1, further comprising: processing the first signal to establishwireless data communications between components of the first wirelesscommunication system; and processing the second signal to at least oneof detect objects and process a parameter or establish wireless datacommunications between components of the second wireless communicationsystem.
 11. A non-transitory computer-readable medium havingcomputer-readable instructions stored thereon, which when executed byone or more processors, configure the one or more processors to: receivea mixture signal; determine a cost function associated with the mixturesignal, and determine a first signal and a second signal from themixture signal based on the cost function, the first signal being asignal of a first wireless communication system and the second signalbeing a signal of one of a second wireless communication system or aradar system, the first and second signals being overlappinglytransmitted signals, at least one of the first signal and the secondsignal being used for processing of information associated with the atleast one of the first signal and the second signal.
 12. Thenon-transitory computer-readable medium of claim 11, wherein theoverlapping transmission of the first signal and the second signalincludes at least one of: a spatial overlap of the first signal and thesecond signal; overlapping transmission of the first signal and thesecond signal in a time domain; and overlapping transmission of thefirst signal and the second signal in a frequency domain.
 13. Thenon-transitory computer-readable medium of claim 11, wherein the firstwireless communication system and the second wireless system are basedon different wireless communications standards.
 14. The non-transitorycomputer-readable medium of claim 11, wherein the mixture signal is aconvolutive mixture of a first convolutive process, a second convolutiveprocess and a noise signal, the first convolutive process corresponds toa convolution of the first signal and a first impulse response of thefirst wireless communication system, and the second convolutive processcorresponds to a convolution of the second signal and a second impulseresponse of one of the second wireless communication system or the radarsystem.
 15. The non-transitory computer-readable medium of claim 14,wherein the execution of the computer-readable instructions by the oneor more processors, further configure the one or more processors to:determine the cost function as a function of the first signal and thesecond signal, and minimize the cost function, and determine the firstsignal and the second signal from among possible sets of values of thefirst signal and the second signal that minimize the cost function. 16.The non-transitory computer-readable medium of claim 11, wherein themixture signal is a combination of a convolutive process, a signalrepresented by a combination of the second signal and a transform, and anoise signal, the convolutive process corresponds to a convolution ofthe first signal and an impulse response of the first wirelesscommunication system, and the transform is at least one of anundercomplete, complete and overcomplete transform.
 17. Thenon-transitory computer-readable medium of claim 16, wherein theexecution of the computer-readable instructions by the one or moreprocessors, further configure the one or more processors to: determinethe cost function as a function of the first signal and the secondsignal, and determine the first signal and the second signal from amongpossible sets of values of the first signal and the second signal thatminimize the cost function.
 18. A device comprising: a memory configuredto store computer-readable instructions therein; and one or moreprocessors configured to execute the computer-readable instructions to,receive a mixture signal, and determine a first signal and a secondsignal from the mixture signal, the first signal being a signal of afirst wireless communication system and the second signal being a signalof one of a second wireless communication system or a radar system, thefirst and second signals being overlappingly transmitted signals, atleast one of the first signal and the second signal being used forprocessing of information associated with the at least one of the firstsignal and the second signal.
 19. The device of claim 18, wherein theone or more processors are further configured to execute thecomputer-readable instructions to, determine a cost function associatedwith the mixture signal; and determine the first signal and the secondsignal from among possible sets of values of the first signal and thesecond signal that minimize the cost function.
 20. The device of claim18, wherein the overlapping transmission of the first signal and thesecond signal includes at least one of: a spatial overlap of the firstsignal and the second signal; overlapping transmission of the firstsignal and the second signal in a time domain; and overlappingtransmission of the first signal and the second signal in a frequencydomain; and the first wireless communication system and the secondwireless system are based on different wireless communicationsstandards.